- Email: [email protected]
Auto-balancing of a rotor with an orthotropic elastic shaft
Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Auto-balancing of a rotor with an orthotropic elastic shaft夽 B.G. Bykov St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 16 September 2011
a b s t r a c t The self-balancing of a statically unbalanced orthotropic elastic rotor equipped with a ball auto-balancing device is investigated. Equations of motion in fixed and rotating systems of coordinates, as well as equations describing steady motions of the regular precession type, are derived using a simple model of a Jeffcott rotor. Formulae for calculating the amplitude-frequency and phase-frequency characteristics of the precessional motion of the rotor are obtained. It is established that the conditions for a steady balanced mode of motion for an orthotropic rotor to exist have the same form as for an isotropic rotor, but the stability region of such a mode for an orthotropic rotor is narrower than the stability region for an isotropic rotor. The unsteady modes of motion of the rotor in the case of rotation with constant angular velocity and in the case of passage through critical velocities with constant angular acceleration is investigated numerically. It is established that the mode of slow passage through the critical region for an orthotropic rotor is far more dangerous than the similar mode for an isotropic rotor. © 2013 Elsevier Ltd. All rights reserved.
Ball auto-balancing devices (ABD), intended for achieving complete balancing of high-revolution rotors with variable imbalance, are widely employed in industrial machinery, means of transportation, home appliances, and precision machinery. Despite the fact that the first patent on a ball auto-balancing device appeared more than 120 years ago,1 and the first theoretical study was published more than 80 years ago,2 interest in this subject has not waned, as is indicated both by the numerous articles and patents and by the proprietary technologies that have appeared in the last few years for using ABDs in various fields of mechanical engineering, computer technology and home appliances. A very simple mechanical model in the form of a point mass attached to a weightless shaft (the Jeffcott rotor model3 ) is useful for studying the precessional motions of statically unbalanced rotors. Two versions of the model in the form of a rigid shaft in elastic bearings or a flexible shaft in pivoting bearings are usually considered. In the case of isotropic elastic bearings and an isotropic elastic shaft, the two versions are equivalent. When the anisotropic characteristics of the rotor and the bearings are taken into account, differences appear in the equations that express specific features of the two versions. There are several papers4–8 devoted to features of the auto-balancing of rigid rotors mounted in anisotropic elastic bearings, but the auto-balancing of an anisotropic flexible rotor is not reflected in the literature. 1. Equations of motion Consider a dynamically symmetrical, statically unbalanced rotor in the form of a sturdy thin disk attached to the middle of a weightless orthotropic shaft rotating in vertical pivoting bearings O1 and O2 . To compensate for the static imbalance (which is not necessarily constant), the rotor is equipped with a ball auto-balancing device (ABD) in the form of a circular race, in which n balls of identical mass can move freely. The race is mounted on the same axis as the rotor so that the distances between the centres of the balancing balls and the centre of the disk are identical and equal to r (Fig. 1). We will examine the motion of a rotor within the Jeffcott model only in the plane of static eccentricity, i.e., the horizontal plane passing through the geometric centre of the disk C and the centre of mass G. We will treat the balancing balls as point masses. We introduce three systems of coordinates: a fixed system of coordinates OXYZ, a rotating system of coordinates O and a system of coordinates C , rigidly connected to the rotor. We direct the Z axis of the fixed system vertically upward along the axis connecting the centres of the bearings, and we choose the origin of coordinates so that the X and Y axes lie in the plane of static eccentricity. The axis of the rotating system coincides with the Z axis, and the and axes are collinear with the and axes of the system of coordinates
夽 Prikl. Mat. Mekh. Vol. 77, No. 4, pp. 514–527, 2013. E-mail address: [email protected] 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.11.005
370
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 1.
connected to the rotor. By virtue of the assumption that the elastic properties of the shaft are orthotropic, its stiffness diagram has two ellipses. We direct the axes of the system of coordinates connected to the rotor along the axes of the stiffness ellipse. The coefficients of elasticity of the shaft corresponding to these axes are denoted by k and k . The position of the centre of mass of the disk is specified by the eccentricity s = CG and the phase angle ␣ between the imbalance vector and the axis. By virtue of the assumptions made, the mechanical system described has n + 3 degrees of freedom. We choose the following as generalized coordinates: the coordinates X and Y of the point C in the fixed system, the angle between the OX and O axes (the angle of rotation of the rotor), and the angles of deflection i (i = 1, . . ., n) of the balls relative to the disk. The expressions for the kinetic energy and potential energy of the system have the form
where
m0 is the mass of the rotor, m1 is the mass of a balancing ball, and IG is the moment of inertia of the rotor about the axis passing through the point G perpendicularly to the XY plane. Here and below, unless otherwise stated, the summation is carried out from i = 1 to i = n. Assuming that only external damping forces act on the rotor, we write the expression for the Rayleigh dissipation function
where c0 is the external damping factor and c1 is the coefficient of viscous resistance to the motion of the balls in the race of the ABD. Assuming that the angle = (t) is a specified function of time, we introduce the new variable i = + i and write Lagrange’s equations of the second kind in the generalized coordinates X, Y and i
(1.1) In the case when the rotor rotates with constant angular velocity ˙ = , it is convenient to write the equations of motion in the rotating system of coordinates O. The relation between the coordinates X, Y and , is specified by the orthogonal transformation formulae
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
371
We multiply the first equation in system (1.1) by cos t and sum it with the second equation multiplied by sin t. We then multiply the second equation by cos t and subtract the first equation multiplied by sin t. As a result, we obtain autonomous equations of motion in the variables , and i . Changing to the dimensionless coordinates
we represent the autonomous equations in the form
(1.2) In system (1.2) the differentiation is understood to be with respect to the dimensionless time
and = / is the dimensionless angular velocity of the rotor. The parameters = m1 /m0 and = s/r are assumed to be small, and the remaining parameters have the following meanings:
For simplicity, we will henceforth omit the tilde above the dimensionless variables. To find particular solutions of the form
in system (1.2) we will set the values of all the derivatives of the generalized coordinates equal to zero. As a result, we obtain a system of transcendental equations that describes the steady modes of motion of the rotor in the form of a regular circular precession:
(1.3) 2. Steady modes of motion of an orthotropic rotor without an ABD For a rotor without an ABD, i.e., in the case when = 0, from the first two equations of (1.3) we find
(2.1) whence we obtain formulae for calculating the amplitude and phase angle of the circular precession (2.2) Note that despite the occurrence of external damping, the denominator in expressions (2.1) for an orthotropic rotor, unlike an isotropic rotor, can vanish when there are certain relations between the parameters. The values of the parameter corresponding to this match the critical angular velocities at which the amplitude of the circular precession becomes infinite. We introduce the dimensionless coefficient
which characterizes the degree of anisotropy of the elastic characteristics of the rotor. Then, substituting the expressions (2.3)
372
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 2.
into Eqs (2.1), we obtain the equation for finding the critical frequencies
When the condition (2.4) holds, this equation has two real positive roots, which correspond to two critical angular velocities:
(2.5) In Fig. 2 the solid curves show the dependence of the critical frequencies on the anisotropy parameter , calculated from formulae (2.5) for three values of the damping factors ␦0 . The graphs demonstrate the “divergence” of the critical frequencies as || increases. Condition (2.4) can be written in the form
(2.6) ∗
The parameter ␦0 is the “critical damping”, that is, the smallest damping factor for which the denominators in expressions (2.1) do not ∗ vanish. The dependence of ␦0 on is represented by the dashed curve in Fig. 2. The steady amplitude-frequency and phase-frequency characteristics (AFCs and PFCs) of an orthotropic rotor, calculated from formulae ∗ (2.2) for ␦0 = 0.1, are presented in Fig. 3 for a weakly orthotropic rotor (k2 = 0.9k1 or || = 0.05, ␦0 = 0.05) and for a strongly orthotropic ∗ rotor (k2 = 2k1 or || = 0.333, ␦0 = 0.338). For comparison, the AFCs and PFCs of an isotropic rotor (k1 = k2 ) are shown by the dashed curves. In the former case the damping factor ␦0 does not satisfy condition (2.6); therefore, at the critical angular velocity the orthotropic rotor has a finite amplitude for the deviation of the centre of mass. In the latter case condition (2.6) holds, and the amplitudes a at the critical angular velocity tend to infinity. The phenomenon of an unrestricted increase in the amplitude of the precessional motion of an orthotropic rotor in the vicinity of the critical frequencies has already been noted in Ref. 9. The graphs of the PFCs exhibit a self-centring effect: in the supercritical region the phase angle changes by 180 degrees, i.e., the centre of mass G of the disk becomes closer to the point O than to its geometric centre C. 3. Steady modes of motion of an orthotropic rotor with an ABD It is convenient to investigate the steady modes of motion of a rotor with an ABD in a polar system of coordinates. We choose the length −→ a0 and the angle of rotation 0 of the radius vector OC about the O axis as the parameters required. Substituting the expressions (3.1) into the last equation of (1.3), we obtain (3.2) Consider an unbalanced steady mode, under which a0 = / 0. This mode corresponds to the regular precession of the centre of the disk. It follows from equality (3.2) that either i0 = 0 or i0 = 0 + , i.e., all the balancing balls should lie on a straight line passing through the points O and C. We will next consider an ABD with two balancing balls. Then in the case of regular precession only two versions of their mutual arrangement are possible: with 20 = 10 or with 20 = 10 + .
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
373
Fig. 3.
We transform the first two equations of system (1.3), taking replacement (3.1) and equality (3.2) into account. Multiplying the first equation by cos 0 and adding it to the second multiplied by sin 0 and then multiplying the first equation by sin 0 and subtracting the second multiplied by cos 0 , we obtain
(3.3) where
In the case of regular precession, the coefficient ␥ can take only one of three possible values: 1, –1 or 0. Each of them corresponds to one of three types of unbalanced steady modes, which we will call modes of types 2+ , 2– and 3 (Ref. 10). Figure 4 shows the arrangement of the balancing balls on the rotor disk for different types of unbalanced steady modes. Note that modes of types 2− and 3 are unstable for any values of the parameters, since under the action of the centrifugal forces the balls tend to occupy the position most distant from the point O on the disk. The stability of an unbalanced steady mode with an ABD was previously investigated for an isotropic rotor.1 We will examine mode 2+ and construct the AFC and PFC for it. Expressing cos2 0 and sin2 0 in terms of cos 20 in system (3.3) and taking into account that
Fig. 4.
374
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 5.
we obtain the equations
(3.4) After multiplying the second of them by imaginary unit i and summing with the first, we arrive at a single complex equation in a0 , o and (3.5) Treating this equation as a quadratic equation in
ei0
and solving it, we find
whence we obtain the equation in a0 and (3.6) which is suitable for calculating the AFC of the rotor. After dividing the first equation of (3.4) by the second, we obtain the equation for calculating the PFC
(3.7) We introduce the balancing coefficient
which expresses the ratio of the maximum total momentum created by the balancing balls to the magnitude of the imbalance vector. Figure 5 presents the AFCs and PFCs of the steady modes of type 2+ of a highly orthotropic rotor (|| = 0.333), which were calculated for the following values of the dimensionless parameters
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
375
and two values of the balancing coefficient . The graphs demonstrate that in the case when = 0.8, mode 2+ exists over the entire range of frequencies and that in the case when = 1.2 it does not exist in the supercritical region. Setting 0 = 0 = 0 in Eqs (1.3), we obtain the equations
(3.8) which describe a balanced steady mode. They are distinguished from the similar equations for an isotropic rotor11 only by the presence of the phase angle ␣. In the case of an ABD with one ball, i.e., when n = 1, system (3.8) has the unique solution * = + ␣, which exists only when the condition = holds. Therefore, one ball is insufficient for auto-balancing of a rotor with variable imbalance. When n = 2 and the condition ≥ /2 or the condition ≥ 1 holds, system (3.8) has the following solution, which corresponds to a balanced mode: (3.9) It follows from this solution that the two balancing balls occupy symmetrical positions relative to the centre of mass of the disk, for which the centre of mass of the entire system is located at the point C.
4. Stability of the balanced steady mode We will investigate the stability of the balanced steady mode in a first approximation. Suppose , and i (i = 1, 2) are small deviations of the generalized coordinates from the stationary values corresponding to the balanced mode. Substituting the expressions
into Eqs (1.2), expanding in a series in the small deviations and neglecting the small deviations of the second order and higher, we obtain a linear system of variational equations, which we write in matrix form as the linear eighth-order system (4.1) where
E is the unit matrix, and O is a 4 × 4 zero matrix.
376
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
We write expressions for the coefficients of the characteristic polynomial of system (4.1), taking relations (2.3) and (3.9) into account
where
A necessary condition for stability of the balanced steady mode, which follows from the condition that the coefficient a7 is positive, has the form (4.2) For an isotropic rotor (i.e., in the case when = 0) condition (4.2) is identical with the previously obtained condition.11 A necessary and sufficient condition for the real parts of the roots of the characteristic equation to be negative is that the Routh coefficients ci,1 (i = 1, . . ., 9) are positive when they are calculated from the recursion formulae
where
The calculations performed with
specify the following regions of asymptotic stability of the balanced steady mode: for an isotropic rotor ( = 0) > 1.024, for an orthotropic rotor (|| = 0.333, ␣ = 0.25) > 1.177. It is convenient to evaluate the influence of the individual parameters of the system using two-parameter stability diagrams. Figure 6 presents stability diagrams in the (, ␦1 ) and (, ␦0 ) parameter planes, which were calculated using the Routh criteria for isotropic and orthotropic rotors. The left-hand part corresponds to the case when ␦0 = 0.05, and the middle part corresponds to the case when ␦1 = 10. The stability regions of the balanced steady mode for the orthotropic rotor are shown hatched. The boundaries of the stability region for the isotropic rotor are marked by dashed lines. The right-hand part demonstrates the narrowing of the stability region in the (, ) parameter plane as the anisotropy parameter increases.
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
377
Fig. 6.
5. Transitional modes Unsteady modes of motion of an isotropic rotor equipped with an ABD were considered previously.12 The behaviour of an anisotropic rotor during the transition to a steady mode was investigated by numerically integrating system (2.1) for constant values of the angular velocity . The results of the calculations are shown in Fig. 7, where the solid curves correspond to the orthotropic rotor (|| = 0.333) and the dashed curves correspond to the isotropic rotor. The left-hand part presents graphs of the time dependence of the displacement of
X 2 + Y 2 and of the angles of deflection of the balancing balls 1 and 2 , calculated in the subcritical region the centre of the disk a = ( < 1), and the middle and right-hand parts show the similar results in the supercritical region ( > 10). It is seen from the graphs that when = 0.75, the radius of circular precession of the anisotropic rotor is greater than that of the isotropic rotor. When = 1.2, anisotropy of the shaft results in an increase in the duration of the transitional process, and when = 1.5, anisotropy has virtually no influence on the motion of the rotor and the balancing balls. Modes with unsteady passage through the critical region, which appear when the rotor accelerates from a state of rest or in the case of stopping, when its angular velocity is above the critical value, are of special interest for practical purposes. Figure 8 presents the results of calculations for a rotor rotating with a constant angular acceleration ¨ = 2 = const, which were obtained by numerical integration of system (1.1). The left-hand part of the figure corresponds to a mode of “slow” passage ( = 0.02) through the critical region, and the right-hand part corresponds to “fast” passage ( = 0.08). It is seen that in the case of slow passage through the critical region the maximum deviation of the centre of the anisotropic rotor (the solid curve) is several times greater than the maximum deviation of the rotor with an isotropic shaft (the dashed curve). When passage through the critical region is faster, no strong discrepancy between the deviations for the isotropic and orthotropic rotors is observed.
Fig. 7.
378
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 8.
6. Conclusions Based on the foregoing, we can draw the following conclusions regarding the influence of the orthotropy of the elastic properties of a flexible shaft on the motion of a rotor equipped with an ABD. 1. The orthotropy of the flexible shaft results in the appearance of a second critical frequency and is also the cause (when the external damping is less than the critical damping) of an unrestricted increase in the amplitude of the precessional motion. 2. The conditions for the existence of a balanced steady mode for an orthotropic rotor have the same form as those for an isotropic rotor, but the stability region for a given mode narrows as the anisotropy parameter increases. 3. A numerical investigation of transitional modes of motion shows that when the rotor rotates with a constant angular velocity, the anisotropy has an appreciable influence on the motion of the rotor in the subcritical region and near the critical frequencies. In the supercritical region, sufficiently far from the critical frequencies the orthotropy of the shaft does not have a significant influence on the auto-balancing process. 4. During slow unsteady passage through the critical region, the deviation of the centre of the disk in the case of an orthotropic shaft can be several times greater than the similar deviation in the case of a rotor with an isotropic shaft. References 1. 2. 3. 4. 5. 6. 7. 8.
Herrick GM. Self-adjusting counter-balance. USA Patent No. 414642. Published 5 November 1889. Thearle EL. A new type of dynamic-balancing machine. Trans ASME 1932;54(12):131–41. Genta G. Dynamics of Rotating Systems. New York: Springer; 2005. Agafonov YuV, Bazykin YuV. Investigation of the stability of a ball auto-balancer of a rotor system on anisotropic bearings. Mashinovedeniye 1985;(5):111–3. Nesterenko VP. Automatic elimination of the static imbalance of a rotor with anisotropic bearings. Mashinovedeniye 1984;(1):24–5. Olsson KO. Limits for the use of auto-balancing. Int J Rotating Machinery 2004;10(3):221–6. Ryzhik B, Sperling L, Duckstein H. Auto-balancing of anisotropically supported rigid rotors. Techn Mech 2004;(24):37–50. Rodrigues DJ, Champneys AR, Friswell MI, Wilson RE. Two-plane automatic balancing: A symmetry breaking analysis. Int J Non Linear Mech 2011;46(9): 1139–54.
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
379
9. Dimentberg FM, Shatalov KT, Gusarov AA. Vibrations of Machines. Moscow: Mashinostroyeniye; 1964. 10. Green K, Champneys AR, Lieven NJ. Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors. J Sound Vib 2006;291: 861–81. 11. Bykov VG. Unsteady modes of motion of an unbalanced rotor with an auto-balancing mechanism. Vestnik SPb Univ Ser 2006;1(2):90–102. 12. Bykov VG. Unsteady modes of motion of a statically unbalanced rotor with an auto-balancing mechanism. Vestnik SPb Univ Ser 2010;1(3): 89–96.
Translated by P.S.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
Auto-balancing of a rotor with an orthotropic elastic shaft夽 B.G. Bykov St Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 16 September 2011
a b s t r a c t The self-balancing of a statically unbalanced orthotropic elastic rotor equipped with a ball auto-balancing device is investigated. Equations of motion in fixed and rotating systems of coordinates, as well as equations describing steady motions of the regular precession type, are derived using a simple model of a Jeffcott rotor. Formulae for calculating the amplitude-frequency and phase-frequency characteristics of the precessional motion of the rotor are obtained. It is established that the conditions for a steady balanced mode of motion for an orthotropic rotor to exist have the same form as for an isotropic rotor, but the stability region of such a mode for an orthotropic rotor is narrower than the stability region for an isotropic rotor. The unsteady modes of motion of the rotor in the case of rotation with constant angular velocity and in the case of passage through critical velocities with constant angular acceleration is investigated numerically. It is established that the mode of slow passage through the critical region for an orthotropic rotor is far more dangerous than the similar mode for an isotropic rotor. © 2013 Elsevier Ltd. All rights reserved.
Ball auto-balancing devices (ABD), intended for achieving complete balancing of high-revolution rotors with variable imbalance, are widely employed in industrial machinery, means of transportation, home appliances, and precision machinery. Despite the fact that the first patent on a ball auto-balancing device appeared more than 120 years ago,1 and the first theoretical study was published more than 80 years ago,2 interest in this subject has not waned, as is indicated both by the numerous articles and patents and by the proprietary technologies that have appeared in the last few years for using ABDs in various fields of mechanical engineering, computer technology and home appliances. A very simple mechanical model in the form of a point mass attached to a weightless shaft (the Jeffcott rotor model3 ) is useful for studying the precessional motions of statically unbalanced rotors. Two versions of the model in the form of a rigid shaft in elastic bearings or a flexible shaft in pivoting bearings are usually considered. In the case of isotropic elastic bearings and an isotropic elastic shaft, the two versions are equivalent. When the anisotropic characteristics of the rotor and the bearings are taken into account, differences appear in the equations that express specific features of the two versions. There are several papers4–8 devoted to features of the auto-balancing of rigid rotors mounted in anisotropic elastic bearings, but the auto-balancing of an anisotropic flexible rotor is not reflected in the literature. 1. Equations of motion Consider a dynamically symmetrical, statically unbalanced rotor in the form of a sturdy thin disk attached to the middle of a weightless orthotropic shaft rotating in vertical pivoting bearings O1 and O2 . To compensate for the static imbalance (which is not necessarily constant), the rotor is equipped with a ball auto-balancing device (ABD) in the form of a circular race, in which n balls of identical mass can move freely. The race is mounted on the same axis as the rotor so that the distances between the centres of the balancing balls and the centre of the disk are identical and equal to r (Fig. 1). We will examine the motion of a rotor within the Jeffcott model only in the plane of static eccentricity, i.e., the horizontal plane passing through the geometric centre of the disk C and the centre of mass G. We will treat the balancing balls as point masses. We introduce three systems of coordinates: a fixed system of coordinates OXYZ, a rotating system of coordinates O and a system of coordinates C , rigidly connected to the rotor. We direct the Z axis of the fixed system vertically upward along the axis connecting the centres of the bearings, and we choose the origin of coordinates so that the X and Y axes lie in the plane of static eccentricity. The axis of the rotating system coincides with the Z axis, and the and axes are collinear with the and axes of the system of coordinates
夽 Prikl. Mat. Mekh. Vol. 77, No. 4, pp. 514–527, 2013. E-mail address: [email protected] 0021-8928/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2013.11.005
370
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 1.
connected to the rotor. By virtue of the assumption that the elastic properties of the shaft are orthotropic, its stiffness diagram has two ellipses. We direct the axes of the system of coordinates connected to the rotor along the axes of the stiffness ellipse. The coefficients of elasticity of the shaft corresponding to these axes are denoted by k and k . The position of the centre of mass of the disk is specified by the eccentricity s = CG and the phase angle ␣ between the imbalance vector and the axis. By virtue of the assumptions made, the mechanical system described has n + 3 degrees of freedom. We choose the following as generalized coordinates: the coordinates X and Y of the point C in the fixed system, the angle between the OX and O axes (the angle of rotation of the rotor), and the angles of deflection i (i = 1, . . ., n) of the balls relative to the disk. The expressions for the kinetic energy and potential energy of the system have the form
where
m0 is the mass of the rotor, m1 is the mass of a balancing ball, and IG is the moment of inertia of the rotor about the axis passing through the point G perpendicularly to the XY plane. Here and below, unless otherwise stated, the summation is carried out from i = 1 to i = n. Assuming that only external damping forces act on the rotor, we write the expression for the Rayleigh dissipation function
where c0 is the external damping factor and c1 is the coefficient of viscous resistance to the motion of the balls in the race of the ABD. Assuming that the angle = (t) is a specified function of time, we introduce the new variable i = + i and write Lagrange’s equations of the second kind in the generalized coordinates X, Y and i
(1.1) In the case when the rotor rotates with constant angular velocity ˙ = , it is convenient to write the equations of motion in the rotating system of coordinates O. The relation between the coordinates X, Y and , is specified by the orthogonal transformation formulae
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
371
We multiply the first equation in system (1.1) by cos t and sum it with the second equation multiplied by sin t. We then multiply the second equation by cos t and subtract the first equation multiplied by sin t. As a result, we obtain autonomous equations of motion in the variables , and i . Changing to the dimensionless coordinates
we represent the autonomous equations in the form
(1.2) In system (1.2) the differentiation is understood to be with respect to the dimensionless time
and = / is the dimensionless angular velocity of the rotor. The parameters = m1 /m0 and = s/r are assumed to be small, and the remaining parameters have the following meanings:
For simplicity, we will henceforth omit the tilde above the dimensionless variables. To find particular solutions of the form
in system (1.2) we will set the values of all the derivatives of the generalized coordinates equal to zero. As a result, we obtain a system of transcendental equations that describes the steady modes of motion of the rotor in the form of a regular circular precession:
(1.3) 2. Steady modes of motion of an orthotropic rotor without an ABD For a rotor without an ABD, i.e., in the case when = 0, from the first two equations of (1.3) we find
(2.1) whence we obtain formulae for calculating the amplitude and phase angle of the circular precession (2.2) Note that despite the occurrence of external damping, the denominator in expressions (2.1) for an orthotropic rotor, unlike an isotropic rotor, can vanish when there are certain relations between the parameters. The values of the parameter corresponding to this match the critical angular velocities at which the amplitude of the circular precession becomes infinite. We introduce the dimensionless coefficient
which characterizes the degree of anisotropy of the elastic characteristics of the rotor. Then, substituting the expressions (2.3)
372
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 2.
into Eqs (2.1), we obtain the equation for finding the critical frequencies
When the condition (2.4) holds, this equation has two real positive roots, which correspond to two critical angular velocities:
(2.5) In Fig. 2 the solid curves show the dependence of the critical frequencies on the anisotropy parameter , calculated from formulae (2.5) for three values of the damping factors ␦0 . The graphs demonstrate the “divergence” of the critical frequencies as || increases. Condition (2.4) can be written in the form
(2.6) ∗
The parameter ␦0 is the “critical damping”, that is, the smallest damping factor for which the denominators in expressions (2.1) do not ∗ vanish. The dependence of ␦0 on is represented by the dashed curve in Fig. 2. The steady amplitude-frequency and phase-frequency characteristics (AFCs and PFCs) of an orthotropic rotor, calculated from formulae ∗ (2.2) for ␦0 = 0.1, are presented in Fig. 3 for a weakly orthotropic rotor (k2 = 0.9k1 or || = 0.05, ␦0 = 0.05) and for a strongly orthotropic ∗ rotor (k2 = 2k1 or || = 0.333, ␦0 = 0.338). For comparison, the AFCs and PFCs of an isotropic rotor (k1 = k2 ) are shown by the dashed curves. In the former case the damping factor ␦0 does not satisfy condition (2.6); therefore, at the critical angular velocity the orthotropic rotor has a finite amplitude for the deviation of the centre of mass. In the latter case condition (2.6) holds, and the amplitudes a at the critical angular velocity tend to infinity. The phenomenon of an unrestricted increase in the amplitude of the precessional motion of an orthotropic rotor in the vicinity of the critical frequencies has already been noted in Ref. 9. The graphs of the PFCs exhibit a self-centring effect: in the supercritical region the phase angle changes by 180 degrees, i.e., the centre of mass G of the disk becomes closer to the point O than to its geometric centre C. 3. Steady modes of motion of an orthotropic rotor with an ABD It is convenient to investigate the steady modes of motion of a rotor with an ABD in a polar system of coordinates. We choose the length −→ a0 and the angle of rotation 0 of the radius vector OC about the O axis as the parameters required. Substituting the expressions (3.1) into the last equation of (1.3), we obtain (3.2) Consider an unbalanced steady mode, under which a0 = / 0. This mode corresponds to the regular precession of the centre of the disk. It follows from equality (3.2) that either i0 = 0 or i0 = 0 + , i.e., all the balancing balls should lie on a straight line passing through the points O and C. We will next consider an ABD with two balancing balls. Then in the case of regular precession only two versions of their mutual arrangement are possible: with 20 = 10 or with 20 = 10 + .
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
373
Fig. 3.
We transform the first two equations of system (1.3), taking replacement (3.1) and equality (3.2) into account. Multiplying the first equation by cos 0 and adding it to the second multiplied by sin 0 and then multiplying the first equation by sin 0 and subtracting the second multiplied by cos 0 , we obtain
(3.3) where
In the case of regular precession, the coefficient ␥ can take only one of three possible values: 1, –1 or 0. Each of them corresponds to one of three types of unbalanced steady modes, which we will call modes of types 2+ , 2– and 3 (Ref. 10). Figure 4 shows the arrangement of the balancing balls on the rotor disk for different types of unbalanced steady modes. Note that modes of types 2− and 3 are unstable for any values of the parameters, since under the action of the centrifugal forces the balls tend to occupy the position most distant from the point O on the disk. The stability of an unbalanced steady mode with an ABD was previously investigated for an isotropic rotor.1 We will examine mode 2+ and construct the AFC and PFC for it. Expressing cos2 0 and sin2 0 in terms of cos 20 in system (3.3) and taking into account that
Fig. 4.
374
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 5.
we obtain the equations
(3.4) After multiplying the second of them by imaginary unit i and summing with the first, we arrive at a single complex equation in a0 , o and (3.5) Treating this equation as a quadratic equation in
ei0
and solving it, we find
whence we obtain the equation in a0 and (3.6) which is suitable for calculating the AFC of the rotor. After dividing the first equation of (3.4) by the second, we obtain the equation for calculating the PFC
(3.7) We introduce the balancing coefficient
which expresses the ratio of the maximum total momentum created by the balancing balls to the magnitude of the imbalance vector. Figure 5 presents the AFCs and PFCs of the steady modes of type 2+ of a highly orthotropic rotor (|| = 0.333), which were calculated for the following values of the dimensionless parameters
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
375
and two values of the balancing coefficient . The graphs demonstrate that in the case when = 0.8, mode 2+ exists over the entire range of frequencies and that in the case when = 1.2 it does not exist in the supercritical region. Setting 0 = 0 = 0 in Eqs (1.3), we obtain the equations
(3.8) which describe a balanced steady mode. They are distinguished from the similar equations for an isotropic rotor11 only by the presence of the phase angle ␣. In the case of an ABD with one ball, i.e., when n = 1, system (3.8) has the unique solution * = + ␣, which exists only when the condition = holds. Therefore, one ball is insufficient for auto-balancing of a rotor with variable imbalance. When n = 2 and the condition ≥ /2 or the condition ≥ 1 holds, system (3.8) has the following solution, which corresponds to a balanced mode: (3.9) It follows from this solution that the two balancing balls occupy symmetrical positions relative to the centre of mass of the disk, for which the centre of mass of the entire system is located at the point C.
4. Stability of the balanced steady mode We will investigate the stability of the balanced steady mode in a first approximation. Suppose , and i (i = 1, 2) are small deviations of the generalized coordinates from the stationary values corresponding to the balanced mode. Substituting the expressions
into Eqs (1.2), expanding in a series in the small deviations and neglecting the small deviations of the second order and higher, we obtain a linear system of variational equations, which we write in matrix form as the linear eighth-order system (4.1) where
E is the unit matrix, and O is a 4 × 4 zero matrix.
376
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
We write expressions for the coefficients of the characteristic polynomial of system (4.1), taking relations (2.3) and (3.9) into account
where
A necessary condition for stability of the balanced steady mode, which follows from the condition that the coefficient a7 is positive, has the form (4.2) For an isotropic rotor (i.e., in the case when = 0) condition (4.2) is identical with the previously obtained condition.11 A necessary and sufficient condition for the real parts of the roots of the characteristic equation to be negative is that the Routh coefficients ci,1 (i = 1, . . ., 9) are positive when they are calculated from the recursion formulae
where
The calculations performed with
specify the following regions of asymptotic stability of the balanced steady mode: for an isotropic rotor ( = 0) > 1.024, for an orthotropic rotor (|| = 0.333, ␣ = 0.25) > 1.177. It is convenient to evaluate the influence of the individual parameters of the system using two-parameter stability diagrams. Figure 6 presents stability diagrams in the (, ␦1 ) and (, ␦0 ) parameter planes, which were calculated using the Routh criteria for isotropic and orthotropic rotors. The left-hand part corresponds to the case when ␦0 = 0.05, and the middle part corresponds to the case when ␦1 = 10. The stability regions of the balanced steady mode for the orthotropic rotor are shown hatched. The boundaries of the stability region for the isotropic rotor are marked by dashed lines. The right-hand part demonstrates the narrowing of the stability region in the (, ) parameter plane as the anisotropy parameter increases.
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
377
Fig. 6.
5. Transitional modes Unsteady modes of motion of an isotropic rotor equipped with an ABD were considered previously.12 The behaviour of an anisotropic rotor during the transition to a steady mode was investigated by numerically integrating system (2.1) for constant values of the angular velocity . The results of the calculations are shown in Fig. 7, where the solid curves correspond to the orthotropic rotor (|| = 0.333) and the dashed curves correspond to the isotropic rotor. The left-hand part presents graphs of the time dependence of the displacement of
X 2 + Y 2 and of the angles of deflection of the balancing balls 1 and 2 , calculated in the subcritical region the centre of the disk a = ( < 1), and the middle and right-hand parts show the similar results in the supercritical region ( > 10). It is seen from the graphs that when = 0.75, the radius of circular precession of the anisotropic rotor is greater than that of the isotropic rotor. When = 1.2, anisotropy of the shaft results in an increase in the duration of the transitional process, and when = 1.5, anisotropy has virtually no influence on the motion of the rotor and the balancing balls. Modes with unsteady passage through the critical region, which appear when the rotor accelerates from a state of rest or in the case of stopping, when its angular velocity is above the critical value, are of special interest for practical purposes. Figure 8 presents the results of calculations for a rotor rotating with a constant angular acceleration ¨ = 2 = const, which were obtained by numerical integration of system (1.1). The left-hand part of the figure corresponds to a mode of “slow” passage ( = 0.02) through the critical region, and the right-hand part corresponds to “fast” passage ( = 0.08). It is seen that in the case of slow passage through the critical region the maximum deviation of the centre of the anisotropic rotor (the solid curve) is several times greater than the maximum deviation of the rotor with an isotropic shaft (the dashed curve). When passage through the critical region is faster, no strong discrepancy between the deviations for the isotropic and orthotropic rotors is observed.
Fig. 7.
378
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
Fig. 8.
6. Conclusions Based on the foregoing, we can draw the following conclusions regarding the influence of the orthotropy of the elastic properties of a flexible shaft on the motion of a rotor equipped with an ABD. 1. The orthotropy of the flexible shaft results in the appearance of a second critical frequency and is also the cause (when the external damping is less than the critical damping) of an unrestricted increase in the amplitude of the precessional motion. 2. The conditions for the existence of a balanced steady mode for an orthotropic rotor have the same form as those for an isotropic rotor, but the stability region for a given mode narrows as the anisotropy parameter increases. 3. A numerical investigation of transitional modes of motion shows that when the rotor rotates with a constant angular velocity, the anisotropy has an appreciable influence on the motion of the rotor in the subcritical region and near the critical frequencies. In the supercritical region, sufficiently far from the critical frequencies the orthotropy of the shaft does not have a significant influence on the auto-balancing process. 4. During slow unsteady passage through the critical region, the deviation of the centre of the disk in the case of an orthotropic shaft can be several times greater than the similar deviation in the case of a rotor with an isotropic shaft. References 1. 2. 3. 4. 5. 6. 7. 8.
Herrick GM. Self-adjusting counter-balance. USA Patent No. 414642. Published 5 November 1889. Thearle EL. A new type of dynamic-balancing machine. Trans ASME 1932;54(12):131–41. Genta G. Dynamics of Rotating Systems. New York: Springer; 2005. Agafonov YuV, Bazykin YuV. Investigation of the stability of a ball auto-balancer of a rotor system on anisotropic bearings. Mashinovedeniye 1985;(5):111–3. Nesterenko VP. Automatic elimination of the static imbalance of a rotor with anisotropic bearings. Mashinovedeniye 1984;(1):24–5. Olsson KO. Limits for the use of auto-balancing. Int J Rotating Machinery 2004;10(3):221–6. Ryzhik B, Sperling L, Duckstein H. Auto-balancing of anisotropically supported rigid rotors. Techn Mech 2004;(24):37–50. Rodrigues DJ, Champneys AR, Friswell MI, Wilson RE. Two-plane automatic balancing: A symmetry breaking analysis. Int J Non Linear Mech 2011;46(9): 1139–54.
B.G. Bykov / Journal of Applied Mathematics and Mechanics 77 (2013) 369–379
379
9. Dimentberg FM, Shatalov KT, Gusarov AA. Vibrations of Machines. Moscow: Mashinostroyeniye; 1964. 10. Green K, Champneys AR, Lieven NJ. Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors. J Sound Vib 2006;291: 861–81. 11. Bykov VG. Unsteady modes of motion of an unbalanced rotor with an auto-balancing mechanism. Vestnik SPb Univ Ser 2006;1(2):90–102. 12. Bykov VG. Unsteady modes of motion of a statically unbalanced rotor with an auto-balancing mechanism. Vestnik SPb Univ Ser 2010;1(3): 89–96.
Translated by P.S.